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Thread: Find two positive divergent sequences such that..

  1. #1
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    Find two positive divergent sequences such that..

    Hello
    Find $\displaystyle a_n$ and $\displaystyle b_n$ such that:
    {$\displaystyle an$} , {$\displaystyle b_n$} are divergent sequences
    $\displaystyle a_n,b_n > 0$ for $\displaystyle n\geq1$
    and { $\displaystyle a_n+b_n$ } is a convergent sequences.

    The problem here is both must positive.
    If not, $\displaystyle a_n=n$ and $\displaystyle b_n=-n$ is a nice counter-example.

    But they must be positive.
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    Quote Originally Posted by General View Post
    Hello
    Find $\displaystyle a_n$ and $\displaystyle b_n$ such that:
    {$\displaystyle an$} , {$\displaystyle b_n$} are divergent sequences
    $\displaystyle a_n,b_n > 0$ for $\displaystyle n\geq1$
    and { $\displaystyle a_n+b_n$ } is a convergent sequences.

    The problem here is both must positive.
    If not, $\displaystyle a_n=n$ and $\displaystyle b_n=-n$ is a nice counter-example.

    But they must be positive.

    Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

    Bigger hint;
    Spoiler:
    think sine and cosine.
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    Quote Originally Posted by pomp View Post
    Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

    Bigger hint;
    Spoiler:
    think sine and cosine.
    But sine and cosine are not positive for all $\displaystyle n\geq1$
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    Quote Originally Posted by General View Post
    But sine and cosine are not positive for all $\displaystyle n\geq1$
    Well then do something to them that will ensure they are...
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    You mean $\displaystyle sin^2n$ and $\displaystyle cos^2n$ ?
    What the limit of them ?
    What the limit of thier sum ?
    as $\displaystyle n \rightarrow \infty$
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    Quote Originally Posted by General View Post
    You mean $\displaystyle sin^2n$ and $\displaystyle cos^2n$ ?
    What the limit of them ?
    What the limit of thier sum ?
    as $\displaystyle n \rightarrow \infty$
    "What the limit of them?"

    Neither converge to a limit, they oscillate in (0,1)

    "What is the limit of thier sum? "

    What is $\displaystyle \sin^2 (x) + \cos^2(x)$ ?
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